Question

The area of a circular field is 25 m$^2$. The radius, r, is to be determined using the Newton-Raphson iterative method. For an initial guess of r = 2.500 m, the revised estimate of r after one iteration is \(\underline{\hspace{2cm}}\) m (rounded off to three decimal places).

Hint:

Newton–Raphson works best when the initial guess is close to the actual root.

Answer 1

Correct Answer: 2.83 - 2.85

We want to determine the radius of a circular field whose area is 25 m$^2$. The equation is:
$A = \pi r^2 = 25$
Define the function:
$f(r) = \pi r^2 - 25$
and the derivative:
$f'(r) = 2\pi r$
Using Newton–Raphson formula:
$r_{n+1} = r_n - \dfrac{f(r_n)}{f'(r_n)}$
With $r_0 = 2.500$:
$f(2.5) = \pi(2.5)^2 - 25 = 19.635 - 25 = -5.365$
$f'(2.5) = 2\pi(2.5) = 15.708$
Thus:
$r_1 = 2.5 - \dfrac{-5.365}{15.708} = 2.8415$
Rounded to three decimals: $r_1 = 2.842$ m.